Krieger's type of nonsingular Poisson suspensions and IDPFT systems

TL;DR

This paper constructs explicit examples of sharply weak mixing nonsingular Poisson actions for amenable groups, covering all Krieger types, and introduces new Bernoulli and IDPFT systems with these properties.

Contribution

It provides the first explicit constructions of sharply weak mixing nonsingular Poisson actions of all Krieger types for any amenable group, including new Bernoulli and IDPFT systems.

Findings

Constructed Poisson $ ext{Gamma}$-actions of all Krieger types.

Produced new examples of nonsingular Bernoulli actions.

Developed IDPFT systems with diverse Krieger types.

Abstract

Given an infinite countable discrete amenable group $\Gamma$, we construct explicitly sharply weak mixing nonsingular Poisson $\Gamma$-actions of each Krieger's type: $III_\lambda$, for $\lambda\in[0,1]$, and $II_\infty$. The result is new even for $\Gamma=\Bbb Z$. As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli $\Gamma$-actions and IDPFT systems of each possible Krieger's type.